Phase Transition Without Long-Range Order in Two Dimensions

As a contribution to the problem of sharp phase transitions in such two-dimensional systems for which the absence of long-range order at any finite temperature has been proved, we have investigated the structure function of a harmonic crystal, the conductivity of a two-dimensional superconductor and the magnetic susceptibility for a planar ferromagnet. We use only approximations that satisfy the condition of no long-range order and find that this does not exclude qualitative differences between a quasiordered low temperature phase and a usual disordered high temperature phase. The Bragg scattering peaks of a crystal turn out to be only slightly weakened as compared to the usual delta-function spikes of the structure function. In the frame of the Ginzburg-Landau theory the conductivity of a two-dimensional superconductor exhibits a delta-function singularity at zero frequency that has the same strength as that in the three-dimensional case. Thus we conclude that the absence of long-range order is not sufficient to exclude phase transitions in two-dimensional systems. Finally criteria for two-dimensional behavior of thin slabs of superconducting material are discussed.